Bezier curve not changing animation movement?
Suggestion.

Reduce the number of keyframes.  You have not shown your frames per second.

Consider having a IK target to have a [Follow Path Constraint].  Paths can be infinitely smooth ... only limited by your frames per second. Insert Keyframes beginning with 2 only. Search Follow Path Constraint here at BSE.

See the addon Simplify Curves.

Do one of 2 actions to archive your current keyframes.


Archive your file. Copy the file.
Create a scene which is a copy of the first scene.


Move your keyframes further down the timeline beginning at frame 50. You can also delete the current set of keyframes to live dangerously.

The result should be no keyframes between frame 1 and frame 40.

Next insert 2 keyframes only at frame 1 and frame 40 and judge whether or not it looks choppy.  Consider adding two more  keyframes and again judge whether it looks choppy.  Slowly increase your number of keyframes to see the tension between controlled movement and an excess of keyframes.  If you have both the new and improved keyframes and the old keyframes you can compare them in the playback.  

Only you can judge whether or not something is smooth and fluid, at least with your Blender Animation.  We do not know whether the animation is intended to be karate or the Balshoi Ballet.

Know that IK Inverse Kinematics can sometimes produces a sudden rotation. Poles are meant to control this.